Philosophical aspects of the foundations of mathematics

To complete our initiation to the foundations of mathematics, the following pages (from this one to Concepts of truth), will present an overview of some deeper, more philosophical and intuitive aspects of these foundations (much of which may be already implicitly understood but not well explained by specialists, being not easily seen as proper objects of scientific works). This includes
These things are not necessary for Part 2 (Set Theory, continued) except to explain the deep meaning and consequences of the fact that the set exponentiation or power set (2.7) is not justifiable by the set generation principle. But they will be developed and justified in more details in parts 4 and 5.

1.A. Time in model theory

The time order between interpretations of expressions

Given a model, expressions do not receive their interpretations all at once, but only the ones after the others, because these interpretations depend on each other, thus must be calculated after each other. This time order of interpretation between expressions, follows the hierarchical order from sub-expressions to expressions containing them.
Take for example, the formula xy+x=3. In order for it to make sense, the variables x and y must take a value first. Then, xy takes a value, obtained by multiplying the values of x and y. Then, xy+x takes a value based on the previous ones. Then, the whole formula (xy+x=3) takes a Boolean value (true or false).
But this value depends on those of the free variables x and y. Finally, taking for example the ground formula ∀x, ∃y, xy+x=3, its Boolean value (which is false in the world of real numbers), «is calculated from» those taken by the previous formula for all possible values of x and y, and therefore comes after them.
A finite list of formulas in a theory may be interpreted by a single big formula containing them all. This only requires to successively integrate (or describe) all individual formulas from the list in the big one, with no need to represent formulas as objects (values of a variable). This big formula comes (is interpreted) after them all, but still belongs to the same theory. But for only one formula to describe the interpretation of an infinity of formulas (such as all possible formulas, handled as values of a variable), would require to switch to the framework of one-model theory.

The metaphor of the usual time

I can speak of «what I told about at that time»: it has a sense if that past saying had one, as I got that meaning and I remember it. But mentioning «what I mean», would not itself inform on what it is, as it might be anything, and becomes absurd in a phrase that modifies or contradicts this meaning («the opposite of what I'm saying»). Mentioning «what I will mention tomorrow», even if I knew what I will say, would not suffice to already provide its meaning either: in case I will mention «what I told about yesterday» (thus now) it would make a vicious circle; but even if the form of my future saying ensured that its meaning will exist tomorrow, this would still not provide it today. I might try to speculate on it, but the actual meaning of future statements will only arise once actually expressed in context. By lack of interest to describe phrases without their meaning, we should rather restrict our study to past expressions, while just "living" the present ones and ignoring future ones.
So, my current universe of the past that I can describe today, includes the one of yesterday, but also my yesterday's comments about it and their meaning. I can thus describe today things outside the universe I could describe yesterday. Meanwhile I neither learned to speak Martian nor acquired a new transcendental intelligence, but the same language applies to a broader universe with new objects. As these new objects are of the same kinds as the old ones, my universe of today may look similar to that of yesterday; but from one universe to another, the same expressions can take different meanings.

Like historians, mathematical theories can only «at every given time» describe a universe of past mathematical objects, while this interpretation itself «happens» in a mathematical present outside this universe.
Even if describing «the universe of all mathematical objects» (model of set theory), means describing everything, this «everything» that is described, is only at any time the current universe, the one of our past ; our act of interpreting expressions there, forms our present beyond this past. And then, describing our previous act of description, means adding to this previous description (this «everything» described) something else beyond it.

The infinite time between models

As a one-model theory T' describes a theory T with a model M, the components (notions and structures) of the model [T,M] of T', actually fall into 3 categories: This last part of [T,M] is a mathematical construction determined by the combination of both systems T and M but it is not directly contained in them : it is built after them.
So, the model [T,M] of T', encompassing the present theory T with the interpretation of all its formulas in the present universe M of past objects, is the next universe of the past, which will come as the infinity of all current interpretations (in M) of formulas of T will become past.

Or can it be otherwise ? Would it be possible for a theory T to express or simulate the notion of its own formulas and compute their values ?

Truth undefinability

As we shall see, some theories such as model theory, and set theory from which it can be developed, are actually able to describe themselves: they can describe in each model a system looking like a copy of the same theory, with a notion of "all its formulas" (including objects that are copies of its own formulas). However then, a Truth Undefinability theorem will be established, showing that no single formula (invariant predicate) can ever give the correct boolean values to all object copies of ground formulas, in conformity with the values of these formulas in the same model.

A strong and rigorous proof will be given later. Here is an easy one.

The Berry paradox

This famous paradox is the idea of "defining" a natural number n as "the smallest number not definable in less than twenty words". This would define in 10 words a number... not definable in less than 20 words. But this does not bring a contradiction in mathematics because it is not a mathematical definition. By making it more precise, we can form a simple proof of the truth undefinability theorem (but not a fully rigorous one):
Let us assume a fixed choice of a theory T describing a set ℕ of natural numbers as part of its model M.
Let H be the set of formulas of T with one free variable intended to range over this ℕ, and shorter than (for example) 1000 occurrences of symbols (taken from the finite list of symbols of T, logical symbols and variables).
Consider the formula of T' with one free variable n ranging over ℕ, expressed as
FH, F(n) ⇒ (∃k<n, F(k))
This formula cannot be false on more than one number per formula in H, which are only finitely many (an explicit bound of their number can be found). Thus it must be true on some numbers.
If it was equivalent to some formula BH, we would get

n∈ℕ, B(n) ⇔ (∀FH, F(n) ⇒ (∃k<n, F(k))) ⇒ (∃k<n, B(k))

contradicting the existence of a smallest n on which B is true.
The number 1000 was picked in case translating this formula into T was complicated, ending up in a big formula B, but still in H. If it was so complicated that 1000 symbols didn't suffice, we could try this reasoning starting from a higher number. Since the existence of an equivalent formula in H would anyway lead to a contradiction, no number we might pick can ever suffice to find one. This shows the impossibility to translate such formulas of T' into equivalent formulas of T, by any method much more efficient than the kind of mere enumeration suggested above.
This infinite time between theories, will develop as an endless hierarchy of infinities.

On the incompleteness theorem

(to be completed)
....Provability is expressible as the existence of a number which encodes a proof, made of one existential quantifier that is unbounded in the sense of arithmetic (∃ p, ) where p is an encoding of the proof, and inside is a formula where all quantifiers are bounded, i.e. with finite range (∀x < (...), ...), expressing a verification of this proof.

The time of proving

If no proof of a statement could be found within given limited resources, it may still be a theorem whose proofs could not be found as they may be any longer. This is often unpredictable, for deep theoretical reasons which will appear from the study of the incompleteness theorem and related ones such as Gödel's speed-up theorem :
Set theory and Foundations of Mathematics
1. First foundations of mathematics
1.1. Introduction to the foundations of mathematics
1.2. Variables, sets, functions and operations
1.3. Form of theories: notions,..., meta-objects
1.4. Structures of mathematical systems
1.5. Expressions and definable structures
1.6. Logical connectives
1.7. Classes in set theory
1.8. Binders in set theory
1.9. Axioms and proofs
1.10. Quantifiers
Philosophical aspects
Time in model theory
Set theory as unified framework
2. Set theory - 3. Algebra - 4. Arithmetic 5. Second-order foundations
Other languages:
FR : Temps en théorie des modèles